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Answer:
The given relation R is equivalence relation.
Step-by-step explanation:
Given that:
Where is the relation on the set of ordered pairs of positive integers.
To prove, a relation R to be equivalence relation we need to prove that the relation is reflexive, symmetric and transitive.
1. First of all, let us check reflexive property:
Reflexive property means:
Here we need to prove:
As per the given relation:
which is true.
R is reflexive.
2. Now, let us check symmetric property:
Symmetric property means:
Here we need to prove:
As per the given relation:
means
means
Hence true.
R is symmetric.
3. R to be transitive, we need to prove:
means
.... (1)
means
...... (2)
To prove:
To be we need to prove:
Multiply (1) with (2):
So, R is transitive as well.
Hence proved that R is an equivalence relation.
The sketches are illustrations of inequalities
- <em>The first sketch is: </em>
<em>.</em>
- <em>The second sketch is: </em>
<em />
<em />
<u />
<u>Sketch A</u>
In sketch A, we have the following observations:
- <em>The arrow faces left (this implies less than)</em>
- <em>The dots on -2 and -5 are closed dots (this implies less than or equal to).</em>
<em />
Hence, the inequality is:
<u />
<u>Sketch B</u>
In sketch B, we have the following observations:
- <em>The arrow faces right (this implies greater than)</em>
- <em>The dots on -4 and 3 are closed dots (this implies greater than or equal to).</em>
<em />
Hence, the inequality is:
Read more about inequalities at:
